Due to the 2D nature of the Ising superconductors, the thermal
effect on the induced magnetic orders constitutes an inevitable
issue from both theoretical perspective and experimental
realization. Although the system under study is 2D, the induced
magnetic orders are formed under the combined actions of the
magnetic field, the ISOC and the SC order, so they are not
spontaneous ones. Meanwhile, the induced magnetic orders are
localized inside the vortex core regions, and thus they are local
ones. As a result, one may expect a different manner of the thermal
effect on the induced magnetic orders as compared with the
Mermin-Wagner theorem [1]}. To see the thermal effect on
the induced magnetic orders, we calculate the temperature dependence
of the magnitude of the magnetic orders. Fig. REF (a) shows
the temperature dependence of the maximum of the absolute value for
the induced magnetic order at the vortex core, where \(T\)  is rescaled
by \(T_{c}\approx 0.05\) . As can be seen from the figure, the
magnitude of the magnetic orders remains approximately constant at
low temperature \(T\le 0.02T_{c}\)  [see inset of Fig. REF (a)]
as a result of the small thermal excitations and the almost
unchanged vortex core size at this temperature
regime [2]}. After then it shows a steady decreasing trend
with increasing temperature, and finally reaches a tiny value at
\(T\sim 0.5T_{c}\) . The decreasing trend is mainly ascribed to the
enlarging vortex core size with temperature [2]}, the
so-called Kramer-Pesch Effect [4]}. The enlarged vortex
core would involve more different sublattice sites into the vortex
core center, resulting in the reduction of the induced magnetic
orders. Though the magnetic orders reduce their magnitude upon the
increasing of the temperature, they sustain to a finite temperature.
Therefore, one may expect to observe the induced magnetic orders
under temperatures well below the SC critical temperature.
